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A329864
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Number of compositions of n with the same runs-resistance as cuts-resistance.
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7
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1, 0, 0, 0, 0, 2, 5, 10, 17, 27, 68, 107, 217, 420, 884, 1761, 3679, 7469, 15437, 31396, 64369
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OFFSET
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0,6
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COMMENTS
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A composition of n is a finite sequence of positive integers summing to n.
For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined to be the number of applications required to reach a singleton.
For the operation of shortening all runs by 1, cuts-resistance is defined to be the number of applications required to reach an empty word.
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LINKS
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EXAMPLE
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The a(5) = 2 through a(8) = 17 compositions:
(1112) (1113) (1114) (1115)
(2111) (1122) (1222) (1133)
(2211) (2221) (3311)
(3111) (4111) (5111)
(11211) (11122) (11222)
(11311) (11411)
(21112) (12221)
(22111) (21113)
(111121) (22211)
(121111) (31112)
(111131)
(111221)
(112112)
(112211)
(122111)
(131111)
(211211)
For example, the runs-resistance of (111221) is 3 because we have: (111221) -> (321) -> (111) -> (3), while the cuts-resistance is also 3 because we have: (111221) -> (112) -> (1) -> (), so (111221) is counted under a(8).
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MATHEMATICA
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runsres[q_]:=Length[NestWhileList[Length/@Split[#]&, q, Length[#]>1&]]-1;
degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&, q, Length[#]>0&]]-1;
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], runsres[#]==degdep[#]&]], {n, 0, 10}]
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CROSSREFS
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The version for binary expansion is A329865.
Compositions counted by runs-resistance are A329744.
Compositions counted by cuts-resistance are A329861.
Compositions with runs-resistance = cuts-resistance minus 1 are A329869.
Cf. A003242, A098504, A114901, A242882, A318928, A319411, A319416, A319420, A319421, A329867, A329868.
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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